Question

Consider two normalized eigen functions and , corresponding to the same eigen value. If

Integral a1a2*dT = d

where is real

Find a normalized linear combinations of and that are orthogonal to

(a) a1

(b) a1+a2

Note: The coefficients of the linear combinations need not be real

Answer 1

$\int$ a₁ * a₂ * dT = d;

d/dT = $\int$ a₁ * a₂

d/dT = -a₂;

d/dT = $\int$ (a₁ + a₂) * -(a₂)